Dense baryonic matter in conformally-compensated hidden local symmetry:
Vector manifestation and chiral symmetry restoration
Yong-Liang Ma,1–3,*Masayasu Harada,1,†Hyun Kyu Lee,4,‡Yongseok Oh,3,5,§Byung-Yoon Park,6,∥and Mannque Rho4,7,¶ 1Department of Physics, Nagoya University, Nagoya 464-8602, Japan
2
College of Physics, Jilin University, Changchun 130012, China
3Department of Physics, Kyungpook National University, Daegu 702-701, Korea 4
Department of Physics, Hanyang University, Seoul 133-791, Korea
5Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea 6
Department of Physics, Chungnam National University, Daejeon 305-764, Korea
7Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette cédex, France
(Received 21 May 2014; published 18 August 2014)
We find that, when the dilaton is implemented as a (pseudo-)Nambu-Goldstone boson using a conformal compensator or “conformon” in a hidden gauge symmetric Lagrangian written to Oðp4Þ from which baryons arise as solitons, namely, skyrmions, the vector manifestation and chiral symmetry restoration at high density predicted in hidden local symmetry theory—which is consistent with Brown-Rho scaling— are lost or sent to infinite density. It is shown that they can be restored if in medium the behavior of theω field is taken to deviate from that of theρ meson in such a way that the flavor Uð2Þ symmetry is strongly broken at increasing density. The hitherto unexposed crucial role of the ω meson in the structure of elementary baryon and multibaryon systems is uncovered in this work. In the state of half-skyrmions to which the skyrmions transform at a density n1=2≳ n0 (where n0 is the normal nuclear matter density), characterized by the vanishing (space averaged) quark condensate but nonzero pion decay constant, the nucleon mass remains more or less constant at a value ≳60% of the vacuum value, indicating a large component of the nucleon mass that is not associated with the spontaneous breaking of chiral symmetry. We discuss its connection to the chiral-invariant mass m0that figures in the parity-doublet baryon model. DOI:10.1103/PhysRevD.90.034015 PACS numbers: 12.39.Dc, 12.39.Fe, 21.65.-f, 21.65.Jk
I. INTRODUCTION
The properties of hadronic matter at high density are poorly understood both theoretically and experimentally, and pose a challenge in nuclear and particle physics. They are critically concerned with such issues as the equation of state (EoS) relevant for compact-star matter and the chiral symmetry breaking/restoration in dense matter.
The sign problem in lattice QCD and the nonperturbative nature of the strong interaction in the effective field theory approaches restrict their applicability for dense matter studies. A series of works[1]have shown that the skyrmion approach, where the classical soliton solutions of the mesonic theory capturing quantum chromodynamics (QCD) in the large Nc limit are interpreted as baryons, provides a natural framework for exploring dense baryonic matter without being obstructed by the notorious problems mentioned above. Furthermore, it enables one to study the properties of dense matter and in-medium hadrons in a unified way [2]. However, because of the parameter
dependence in the results of simple models that include only a few low-lying mesons, the skyrmion approach could not provide quantitatively meaningful predictions.
Recently, we have studied the skyrmion approach[3–6]
using a chiral Lagrangian, where theρ and ω mesons are introduced as the gauge bosons of the hidden local symmetry (HLS)[7–9]with the Oðp4Þ terms in the chiral
order expansion including all the homogeneous Wess-Zumino (hWZ) terms taken into account. All the param-eters of the Lagrangian are fixed by their relations to the Sakai-Sugimoto’s five-dimensional holographic QCD (hQCD) model [10], while two parameters of the latter are fixed to yield the empirical values of the pion decay constant and the vector meson mass in matter-free space. We shall refer to this model as HLS(π; ρ; ω).
Several remarkable results, qualitatively different from, or not observed in, the previous works on skyrmions were obtained in Refs.[3–6]. We will return to some of them later. Here we mention a few to illustrate the key issues that will concern what we treat in this paper.
(i) Although the results show some discrepancies from nature, if one considers the fact that none of the parameters is adjusted with baryon properties, they could be taken as the first “parameter-free” predic-tions of the skyrmion approach. Furthermore, in spite of the limitations of the model such as the large Nc,
*[email protected] †[email protected]‑u.ac.jp ‡[email protected] §[email protected] ∥[email protected] ¶[email protected]
large’t Hooft constant (λ), and chiral limits taken in the hQCD model, the semiquantitative agreement of the results with experiments is quite remarkable. (ii) The full Oðp4Þ terms, in particular the hWZ terms
that carry theω meson degree of freedom, are found to be essential for the soliton structure of elementary baryons. This highlights the particularly important role, thus far unexposed, played by theω meson in the baryon structure. We assert that the Lagrangian given up to Oðp4Þ cannot be approximated by a few
terms as was done in the past.1
(iii) In dense matter simulated on the face-centered cubic (FCC) crystal, it is found that the transition from skyrmions to half-skyrmions accompanied by the vanishing quark condensate takes place near the normal nuclear matter density (denoted by n0 from here on) in contrast to the case without the vector mesons and the Oðp4Þ terms. In the latter case, the
transition density, denoted n1=2, is much too high compared with n0. A qualitatively different feature found there, which has not been noticed in the past, is that the medium-modified pion decay constant fπ that decreases smoothly up to n1=2 roughly in the same way as in chiral perturbation calculations stops decreasing at n1=2 and then stays more or less constant, although the space averaged quark con-densationh¯qqi vanishes in the half-skyrmion phase. The nonvanishing fπ in the half-skyrmion phase implies that the chiral symmetry remains still in the Nambu-Goldstone mode even though h¯qqi ¼ 0. Thus, the quark condensate is not a good order parameter in the crystal description.
(iv) It is found that the medium-modified effective nucleon mass mN tracks closely fπ, indicating that the large Nc approximation continues to hold in medium. Furthermore, in the half-skyrmion phase, the mass remains ∼60% of its vacuum value. The condensate-independent mass is reminiscent of the chiral-invariant baryon mass m0 that figures in the parity-doublet baryon model[11], but its physi-cal origin is not clarified yet.
We note that some of the features mentioned above are in disagreement with what was obtained in the mean field approximation[12]and in a naive formulation of HLS in the meson sector[9]. In particular, they bring tension with the vector manifestation (VM)2 and the Brown-Rho (BR) scaling.
Given that what we are dealing with here is an effective theory of QCD given in terms of the“macroscopic” degrees of freedom, i.e., hadrons, we need to match the effective theory to QCD at a certain scale at which the chiral symmetry gets restored. By matching the correlators of HLS to those of QCD in the Wilsonian sense, one finds that near the chiral restoration scale, here the putative critical density nc, the parameters in HLS should satisfy the following relations:
f2πðq2¼ 0Þ → 0; m2ρ→ m2π¼ 0; aðq2¼ m2ρÞ → 1; ð1Þ which are called “vector manifestation” of the Wigner realization of chiral symmetry[13]. The matching between the effective theory and QCD renders the low energy constants (LECs) of the theory intrinsically density dependent.3 However, if, as mentioned, fπ stays as a nonzero constant in the half-skyrmion phase and mρdoes not scale in medium, it is not clear how to access the VM unless some other (hadronic) mechanism intervenes before the QCD degrees of freedom enter.
The objective of this paper is to resolve this problem. For this, we resort to scale (or conformal) symmetry of QCD. It is well known that the chiral symmetry breaking and scale symmetry are intricately linked to each other and that chiral symmetry breaking could be triggered by sponta-neous breaking of scale symmetry which in turn is caused by the presence of explicit scale symmetry breaking
[15,16]. This is the mechanism that we exploit to inves-tigate the possible realization of VM by incorporating the QCD trace anomaly in HLS. We find that, with an appropriate inclusion of the scalar field associated with the explicit breaking of conformal symmetry, both VM and the BR scaling[12]can be realized.
There are a variety of reasons to believe that the scalar degree of freedom is needed in addition to those that figure in HLS in the structure of both elementary baryon and multibaryon systems. Firstly the skyrmion mass obtained in HLS(π; ρ; ω)[3,4]overshoots the empirical nucleon mass by ≳300 MeV. One expects that the OðN0cÞ Casimir
energy, missing in the OðNcÞ soliton mass, can account
for the attraction needed of that amount. In the Skyrme model, the Casimir contribution comes from pion loops that account for the scalar channel and is of the right magnitude to lower the mass from the canonical∼1500 MeV down to
1These terms include the quartic Skyrme term in the
pion-only chiral Lagrangian as in the original Skyrme model or the “minimal” HLS model with one hWZ term out of the three (explained below).
2In the formulation of the VM in Ref.[9], the vanishing quark
condensate (h¯qqi ¼ 0) was used. Of course the vanishing of the ρ meson mass can never occur unless fπ¼ 0 at the chiral restoration point identified with the VM.
3It should be stressed that when we refer to density
depend-ence, we are referring specifically to this intrinsic density dependence [14]. Any truncation in a many-body system in the sense of a Wilsonian renormalization group (RG) would generate density dependence in the parameters of the model, and it would be in practice very difficult to identify or isolate the intrinsic density dependence coming from the matching to QCD in nuclear observables. This has compounded the efforts to see “partial chiral symmetry restoration” from nuclear experiments.
∼1000 MeV[17]. Secondly, in the mean field approach to nuclear matter, a (chiral) scalar mass of ∼600 MeV is indispensable for binding the matter, counterbalancing the repulsion from the ω meson exchange. Closer to the problem at hand, the simulation of the nuclear property from the FCC crystal shows that the density nmin at which
the per-skyrmion energy is the minimum is larger than the normal nuclear density n0and the binding energy at nminis ∼100 MeV, which is larger than the empirical value ∼16 MeV per baryon[5]. These discrepancies also signal that there is something missing in the present skyrmion crystal description for the EoS of nuclear matter in HLS (π; ρ; ω). Although there is such a defect in our approach, in fact, what we are suggesting here is that one can make predictions on certain fluctuation properties of hadrons on top of the skyrmion background which is treated semi-classically, which is what the simulation all about.
The advantage of having the vector mesons as (hidden) gauge fields is that a systematic chiral perturbation theory can be formulated with vector mesons in addition to the pion [5].4 Introducing scalar fields in this framework is, however, highly problematic. What was done in the past and will be done here is to use the trick of the conformal compensator field or“conformon”5used in cosmology and also in the technidilaton approach to a Higgs-like boson, to write a conformally invariant Lagrangian with the con-formal symmetry spontaneously broken by a Coleman-Weinberg-type potential. The conformon field is then identified as a (pseudo-)Nambu-Goldstone boson of spon-taneously broken scale symmetry, i.e., the dilaton.
In what follows we analyze in what way this dilaton resolves the problem mentioned above. This will reveal how it affects the structure of both the elementary nucleon and multibaryon systems at large density.
This paper is organized as follows. In Sec. II, after a sketch of the general strategy of approaching elementary baryon and multibaryon systems with one single Lagrangian anchored on hidden local symmetry, we intro-duce the dilaton associated with the scale symmetry breaking of QCD as a conformon to HLS. This section serves also to define the notations used in the present work. In Sec.IIIthe single skyrmion properties are studied using the dilaton compensated HLS model. The effects of the dilaton on the skyrmion matter properties and medium-modified hadron properties are explored in Sec. IV. A possible way to realize VM and restore chiral symmetry with the effects of dilaton is also discussed. We give a
succinct summary of the results in Sec. V and further discussions in Sec.VI.
II. HIDDEN LOCAL SYMMETRY LAGRANGIAN WITH CONFORMAL INVARIANCE
We start with a brief description of the HLS Lagrangian for defining the notations used in this paper. In free space, the full symmetry group associated with the basic ingre-dients, π, ρ, and ω, is Gfull¼ ½SUð2ÞL× SUð2ÞRchiral×
½Uð2ÞHLS in which the lowest-lying ρ and ω mesons
are incorporated as the gauge bosons of ½SUð2ÞHLS and ½Uð1ÞHLS components, respectively, of the “hidden local
symmetry” ½Uð2ÞHLS. The HLS Lagrangian is constructed
by two 1-forms, ˆα∥μand ˆα⊥μ, defined by
ˆα∥μ¼ 12iðDμξR·ξR† þ DμξL·ξ†LÞ; ð2Þ
ˆα⊥μ ¼ 12iðDμξR·ξR† − DμξL·ξ†LÞ; ð3Þ
with the chiral fieldsξLandξR, which are expressed in the unitary gauge as
ξ†L¼ ξR¼ eiπ=2fπ≡ ξ with π ¼ π · τ; ð4Þ
whereτ’s are the Pauli matrices. The covariant derivative associated with the hidden local symmetry is defined as
DμξR;L ¼ ð∂μ− iVμÞξR;L; ð5Þ
where Vμrepresents the gauge boson of the HLS[7–9]as6 Vμ¼ 1 2ðgωωμþ gρρμÞ ð6Þ and ρμ¼ ρμ·τ ¼ ρ0 μ ffiffiffi 2 p ρþ μ ffiffiffi 2 p ρ− μ −ρ0μ : ð7Þ
Up to Oðp4Þ, including the hWZ terms, the most general
HLS Lagrangian can be expressed as
LHLS¼ LHLSð2Þ þ LHLSð4Þ þ LHLSanom; ð8Þ
with 4The vector manifestation cannot be obtained unless the vector
meson mass can be considered as light as the pion mass as in HLS. The phenomenological Lagrangians used in the literature for treating vector mesons in dense medium can make sense only in mean field and cannot address dropping vector meson masses.
5This term is borrowed from Ref.[18] in which it is used in
cosmology.
6In this paper, we distinguish the gauge coupling constants
for the ω and the ρ mesons which will be convenient for discussing the medium modified hadron properties. In free space, we take the hidden gauge symmetry as Uð2ÞHLS; thus gω¼ gρ≡ g. If Uð2ÞHLSis broken in dense medium, they could have different values.
LHLS
ð2Þ ¼ f2πTrðˆα⊥μˆαμ⊥Þ þ af2πTrðˆα∥μˆαμ∥Þ þ Lkin; ð9Þ
where fπ is the pion decay constant, a is the HLS parameter, andLkincontains the kinetic terms of the vector mesons: Lkin¼ − 1 2g2 ρTrðV ðρÞ μνVðρÞ;μνÞ − 1 2g2 ωTrðV ðωÞ μν VðωÞ;μνÞ; ð10Þ
with the field-strength tensors of vector mesons
VðρÞμν ¼ ∂μ 1 2gρρν − ∂ν 1 2gρρμ − i 1 2gρρμ; 1 2gρρν ; VðωÞμν ¼ ∂μ 1 2gωων − ∂ν 1 2gωωμ : ð11Þ
For later discussions, we have separated the terms for theρ andω mesons to allow different values for gρand gωwhich are the same in the case of½Uð2ÞHLS.
The Oðp4Þ Lagrangian is given by [9,19]
Lð4Þ¼ y1Tr½ˆα⊥μˆαμ⊥ˆα⊥νˆαν⊥ þ y2Tr½ˆα⊥μˆα⊥νˆαμ⊥ˆα⊥ν þ y3Tr½ˆα∥μˆαμ∥ˆα∥νˆαν∥ þ y4Tr½ˆα∥μˆα∥νˆαμ∥ˆαν∥
þ y5Tr½ˆα⊥μˆαμ⊥ˆα∥νˆαν∥ þ y6Tr½ˆα⊥μˆα⊥νˆαμ∥ˆαν∥ þ y7Tr½ˆα⊥μˆα⊥νˆαν∥ˆαμ∥
þ y8fTr½ˆα⊥μˆαμ∥ˆα⊥νˆαν∥ þ Tr½ˆα⊥μˆα∥νˆαν⊥ˆαμ∥g þ y9Tr½ˆα⊥μˆα∥νˆαμ⊥ˆαν∥
þ iz4Tr½VðρÞμνˆαμ⊥ˆαν⊥ þ iz5Tr½VðρÞμν ˆαμ∥ˆαν∥: ð12Þ
Note that VðωÞμν does not appear in the z4 and z5 terms. Finally, the anomalous parity hWZ terms Lanom are
written as ΓhWZ¼ Z d4xLanom ¼ Nc 16π2 Z M4 X3 i¼1 ciLi; ð13Þ where M4 stands for the four-dimensional Minkowski space and
L1¼ iTr½ˆα3LˆαR− ˆα3RˆαL; ð14aÞ
L2¼ iTr½ˆαLˆαRˆαLˆαR; ð14bÞ
L3¼ Tr½FVðˆαLˆαR− ˆαRˆαLÞ; ð14cÞ
in the 1-form and 2-form notations with
ˆαL¼ ˆα∥− ˆα⊥;
ˆαR¼ ˆα∥þ ˆα⊥;
FV¼ dV − iV2: ð15Þ
In the Lagrangian(8) there appear many undetermined constants which include fπ, a, gρ, gω, yiði ¼ 1; …; 9Þ,
ziði ¼ 4; 5Þ, and ciði ¼ 1; 2; 3Þ. To fix them
phenomeno-logically, we need a large number of experimental data, which are not available at present and will not be available in the near future. The recent development of holographic QCD, however, improves the situation dramatically. As discussed in Refs. [3,4], those coefficients can be fixed completely by means of a set of “master formulas” that match the four-dimensional effective theory (here HLS) to
the five-dimensional hQCD model. In the large Nc and
largeλ limit, the hQCD has two parameters which can be related to the empirical values of the pion decay constant and the vector meson mass. Then with these quantities fixed in the meson sector, all the coefficients of the HLS Lagrangian we are dealing with are determined by the master formula. Here we employ the Sakai-Sugimoto hQCD model [10] which is supposed to be dual to our HLS model, with the empirical values
fπ¼ 92.4 MeV; mω¼ mρ¼ 775.5 MeV; ð16Þ where the ½Uð2ÞHLS in free space has been taken.
The essential point in deriving HLS from hQCD models that have 5D Dirac-Born-Infeld part and the Chern-Simons part, S5¼ SDBI 5 þ SCS5 ; ð17Þ where SDBI 5 ¼ NcGYM Z d4xdz −1 2K1ðzÞTr½FμνFμν þ K2ðzÞM2 KKTr½FμzFμz ; ð18Þ SCS 5 ¼ Nc 24π2 Z M4×R w5ðAÞ; ð19Þ
is to make the mode expansion of the 5D gauge field AMðx; zÞ [M ¼ ðμ; zÞ with μ ¼ 0; 1; 2; 3] and integrate
out all the modes except the pseudoscalar and the lowest-lying vector mesons. This reduces AMðx; zÞ to
AintegM ðx; zÞ which in the Azðx; zÞ ¼ 0 gauge amounts to the
substitution[3,4]
Aμðx; zÞ → Aintegμ ðx; zÞ
¼ ˆα⊥μψ0þ ðˆα∥μþ VμÞ þˆα∥μψ1ðzÞ; ð20Þ
where fψnðzÞg are the eigenfunctions satisfying the following eigenvalue equation obtained from the action:
−K−1
1 ðzÞ∂z½K2ðzÞ∂zψnðzÞ ¼ λnψnðzÞ; ð21Þ
withλnbeing the nth eigenvalue (λ0¼ 0). Here, K1ðzÞ and
K2ðzÞ are the warping factors in the fifth direction of the five-dimensional space-time, which are explicitly K1ðzÞ ¼ K−1=3ðzÞ and K2ðzÞ ¼ KðzÞ with KðzÞ ¼ 1 þ z2 in the Sakai-Sugimoto model[10].
In this paper, in order to distinguish theρ and ω mesons associated to the SUð2Þ and Uð1Þ components, respectively, of the 5D gauge field AMðx; zÞ, we rewrite Eq. (20)as
Aμðx; zÞ → Aintegμ ðx; zÞ
¼ ˆα⊥μψ0þ ðˆα∥μþ VμÞ
þˆαSUð2Þ∥μ ψ1ðzÞ þ ˆ~αUð1Þ∥μ ~ψ1ðzÞ; ð22Þ
where again we have separated out the SU(2) part and the U(1) part in the last two terms. The expression for ˆ~αUð1Þ∥μ can be obtained from Eq. (2) by removing all the isotriplets. With these conventions one can easily see that ψ1ðzÞ and ~ψ1ðzÞ are the wave functions of the ρ and ω mesons,
respectively.
Substituting Eq. (22) into the five-dimensional hQCD models leads to the HLS Lagrangian. For the Oðp2Þ terms
we have the following relations for the low energy constants: f2π¼ NcGYMM2KK Z dzK2ðzÞ½_ψ0ðzÞ2; af2π¼ NcGYMM2KKλ1hψ21i; 1 g2ρ¼ NcGYMhψ 2 1i; 1 g2ω ¼ NcGYMh~ψ 2 1i: ð23Þ
The parameters a, fπ, and the gauge coupling constants gρ and gω satisfy the following relations:
ag2ρf2π¼ m2ρ; ag2ωf2π¼ m2ω: ð24Þ Therefore, ψ1ðzÞ and ~ψ1ðzÞ satisfy
~ψ1ðzÞ ¼
mρ
mωψ1ðzÞ: ð25Þ
By using Eq.(25), we get the master formula of the low energy constants of the Oðp4Þ terms as [20]7
y1¼ −y2¼ −f 2 π m2ρNhQCDhð1 þ ψ1− ψ 2 0Þ2i; y3¼ −y4¼ −f 2 π m2ρNhQCDhψ 2 1ð1 þ ψ1Þ2i; y5¼ 2y8¼ −y9¼ −2f 2 π m2ρNhQCDhψ 2 1ψ20i; y6¼ −ðy5þ y7Þ; y7¼ 2f 2 π m2ρNhQCDhψ1ð1 þ ψ1Þð1 þ ψ1− ψ 2 0Þi; z4¼ 2f 2 π m2ρNhQCDhψ1ð1 þ ψ1− ψ 2 0Þi; z5¼ −2f 2 π m2ρNhQCDhψ 2 1ð1 þ ψ1Þi; c1¼mρ mω _ψ0ψ1 1 2ψ20þ 16ψ21−12 ; c2¼mρ mω _ψ0ψ1 −1 2ψ20þ 16ψ21þ 12ψ1þ 12 ; c3¼mρ mω 1 2_ψ0ψ21 ; ð26Þ where NhQCD¼ λ1= R
dzK2ðzÞ½_ψ0ðzÞ2, and the wave func-tion ~ψ1ðzÞ associated with the ω field in ci has been
expressed in terms of ψ1ðzÞ associated with the ρ field through the relation(25). For deriving the expressions of yi and zi, we have considered that the U(1) degree of freedom
should disappear in these terms because of the antisym-metric field tensor appearing in the Dirac-Born-Infeld part. The integrals appearing in the above relations are defined by
hAi ≡ Z ∞ −∞dzK1ðzÞAðzÞ; hhAii ≡ Z ∞ −∞dzAðzÞ: ð27Þ A. Predictions of HLS
To highlight the principal effect of the dilaton in dense skyrmion matter, we briefly review the predictions of dilatonless HLS obtained in the previous works. In Ref. [4] it was shown that this model yields the soliton mass of 1184 MeV, which is quite good as a “parameter-free” result. Although it is larger by about 300 MeV than
7Here, we express the master formula in terms of f
π and mρ.
The factor mρ=mω in ci arises from the normalization of ψ1.
the observed nucleon mass, it is not difficult to understand where this difference may come from. As mentioned in Sec.I, in the standard Skyrme model (with pions only), an excess of∼500 MeV can be reduced by the Casimir energy
[17]that comes at the next order in Nc, i.e., OðN0cÞ. We will see below that the dilaton contributes to remove a part of that excess, although not enough.
The HLS Lagrangian also provides a noticeable improvement in the dense baryonic matter study [5]
compared to what exists in the literature. Here, the skyrmion–half-skyrmion transition takes place near the normal nuclear matter density, rendering the process phenomenologically relevant. A distinctively novel result is that in the half-skyrmion phase, the intrinsic density-dependent (or effective in-medium) pion decay constant fπ is nonvanishing and stays independent of density. The in-medium nucleon mass mN scales similarly to the pion decay constant, which is indicative of the large Nc dominance. This (nearly) constant nucleon mass in the half-skyrmion phase resembles the nonvanishing chiral-invariant mass in the parity-doublet chiral model for baryons[11]. We will return to this matter in the discussion section in connection with the origin of the nucleon mass. What seems not evident is the movement toward the vector manifestation of the HLS given in Eq.(1). In order to arrive at the fixed point that would correspond to the chiral transition, which is a quantum phase transition, the corre-lators of HLS should be matched to those of QCD. For this, it is clear that one has to understand the quantum structure of the half-skyrmion phase. As suggested in Ref.[21], it could involve a topology-triggered change from a Fermi-liquid state to a non-Fermi-Fermi-liquid state. This issue needs to be clarified.
B. Conformally compensating HLS
It is also plausible that higher-order corrections and/or heavier vector mesons such as the a1could play an important role in approaching the chiral restoration point. Our thesis in this paper, as stated in Sec.I, is that what is crucially needed in the HLS structure is the scalar degree of freedom.
As stated in Sec.I, we introduce the scalar needed as a dilaton that figures in spontaneous breaking of scale symmetry (SBSS) which is locked to spontaneous breaking of chiral symmetry (SBCS)[16]. The idea is that the trace anomaly of QCD provides the explicit breaking of scale symmetry that is needed to trigger the SBSS. It is well known that, without the explicit breaking, the spontaneous breaking cannot occur [15]. We associate the part of the gluon condensate that remains “unmelted” above the critical temperature or density, i.e., the “hard glue” in the language of Ref.[14], with the explicit breaking of scale symmetry. We follow the standard procedure of incorpo-rating the nonlinearly realized scale invariance of adding a fieldχ as the “conformal compensator” (or conformon for short). The procedure is to make the HLS Lagrangian
conformally invariant and then add a potential V that breaks conformal invariance spontaneously. The sponta-neous breaking makes the conformon a (pseudo-)Nambu-Goldstone boson, i.e., the dilaton.
If one assumes that the vector fields have scale dimen-sion 1,8then this conformon trick modifies only the Oðp2Þ term in Eq.(8), since the Oðp4Þ terms are scale invariant
as they are. Putting in the dilaton part of the Lagrangian, we have
LdHLS-I¼ LdHLS-Ið2Þ þ LHLSð4Þ þ LHLSanomþ Ldilaton; ð28Þ
where LdHLS-I ð2Þ ¼ f2π χ fχ 2 Tr½ˆα⊥μˆαμ⊥ þ af2π χ fχ 2 Tr½ˆα∥μˆαμ∥ þ Lkin; ð29Þ Ldilaton ¼ 12∂μχ∂μχ þ V: ð30Þ
HereLkin is the kinetic term of vector mesons as given in
Eq.(10)and fχð≠ 0Þ is the vacuum expectation value of the fieldχ. The potential V in this system is not known except that it should reproduce the “soft glue” part in the trace anomaly[14]. If one assumes that the conformal symmetry breaking term is small, then the potential takes the familiar Coleman-Weinberg form (see, e.g., Ref.[22])
V ¼ −m2χ4f2χ χ fχ 4 ln χ fχ −14 þ 1 4 : ð31Þ We shall refer to the Lagrangian (28) with the potential given by Eq.(31) as dHLS-I(π; ρ; ω).
As we shall see below, since the conformon couples in the same way to both theρ and the ω mesons as well as to the derivative of the U field, as the dilaton condensate decreases with density, the energy density of the system diverges with increasing density, as found in the minimal model in Ref.[23].9This leads to a contradictory situation
8This can be supported by the observation that
ρμ∼gi ρð∂μξRξ
†
Rþ ∂μξLξ†LÞ;
in the limit that mρ→ ∞.
9The so-called minimal model corresponds to the truncated
Lagrangian of HLS with the LECs yi¼ zi¼ c3¼ 0 and c1¼
−c2¼ 2=3 in Eq.(8)below. This is gotten by dropping all Oðp4Þ
terms in the Lagrangian except one term∝ ωμBμ(where Bμis the baryon number current) in the hWZ that results if one substitutes the equation of motion for theρ with the ρ mass set to infinity in the hWZ part of the Lagrangian but not elsewhere. Obviously this limit precludes ab initio a droppingρ mass and hence the VM, that we consider unacceptable.
that as the density increases, the pion decay constant and the vector meson mass are forced to increase, instead of decrease.
There are two ways out of this conundrum. Both resort to the possible breaking of ½Uð2ÞHLS symmetry for the ρ andω.
One is to implement the observation made in Ref.[24]
that the symmetry is broken in the gauge coupling gω ≠ gρ. This resolves the above-mentioned problem in a similar way to what is discussed in Ref.[24]. This matter will be addressed in more detail in the discussion section as it raises further issues to be explored.
The other is to break½Uð2ÞHLS symmetry in the mass term so that in medium, while the ρ mass has the VM property, the ω mass does not, which would prevent the increasing repulsion. This can be done by applying the conformal compensator to the ρ sector but not to the ω sector. We have not yet figured out how to justify this structure in a rigorous way.10What we have done in this paper is simply not to apply the conformon to theω mass. To do this we factor out theω mass term in the second term ofLHLSð2Þ and coupleχ2only to the rest. We shall refer to this Lagrangian as dHLS-II(π; ρ; ω). Then it reads
LdHLS-II¼ LdHLS-IIð2Þ þ LHLSð4Þ þ LHLSanomþ Ldilaton; ð32Þ
where LdHLS-II ð2Þ ¼ f2π χ fχ 2 Tr½ˆα⊥μˆαμ⊥ þ af2π χ fχ 2 Tr½ˆα∥μˆαμ∥SUð2Þ þ 1 2af2πg2ωωμωμþ Lkin: ð33Þ
The rest are the same as in dHLS-I(π; ρ; ω). In the second term, the subscript SUð2Þ denotes that only the isovector part is considered and theω mass term is factored out. Note that in free spaceχ=fχ¼ 1, so that the Uð2ÞHLSsymmetry is restored. We are considering the case that the medium breaks½Uð2ÞHLSsymmetry in such a way that, while theρ mass scales in density, the ω mass does not. This is analogous to the weak scaling of theω-nucleon coupling in Ref.[24], where the symmetry breaking is attributed to the vector coupling. We shall see that this simple modi-fication resolves the long-standing problem started in Ref. [23] and enables chiral symmetry to be restored and the vector symmetry to be manifest at some critical density nc.
The incorporation of the dilaton in dHLS-II(π; ρ; ω) brings in two undetermined constants fχand mχ. Since there are no experimental values, we shall just take them as free param-eters. We will present the results obtained with[23]
fχ¼ 240 MeV;
mχ¼ 720 MeV: ð34Þ
In the limit of mχ→ ∞, all of the numerical results trivially converge to those of HLS(π; ρ; ω) reported in Refs.[3–6].
III. SINGLE SKYRMION PROPERTIES IN DHLS-I AND DHLS-II
We first study the effect of the dilaton on the single skyrmion properties. The soliton solution can be found in the spherical form as
ξðrÞ ¼ exp ½iτ · ˆrFðrÞ=2; ρμðrÞ ¼GðrÞ gρr ðˆr × τÞ iδμi; ωμðrÞ ¼ WðrÞδμ0; χðrÞ ¼ fχCðrÞ: ð35Þ
The standard collective rotation quantization [26] is made by the transformation
ξðrÞ → ξðr; tÞ ¼ AðtÞξðrÞA†ðtÞ;
VμðrÞ → Vμðr; tÞ ¼ AðtÞVμðrÞA†ðtÞ; ð36Þ where AðtÞ is a time-dependent SU(2) matrix, which definesΩ by
iτ · Ω ≡ A†ðtÞ∂0AðtÞ; ð37Þ which leads to the most general forms for the vector-meson excitations as ρ0ðr; tÞ ¼ AðtÞ 2 gρ½τ · Ωξ1ðrÞ þˆτ · ˆrΩ · ˆrξ2ðrÞA †ðtÞ; ωiðr; tÞ ¼φðrÞ r ðΩ × ˆrÞ i: ð38Þ
Since the dilaton fieldχ is a spin-0 isoscalar field, it is not affected by the collective rotation. The boundary conditions of the wave functions are given in Ref. [4]and those for CðrÞ read
C0ð0Þ ¼ 0; Cð∞Þ ¼ 1: ð39Þ It is then straightforward to calculate the soliton mass and the moment of inertia from which the equations of motion for the wave functions introduced in Eqs.(35)and
(38)can be read. We refer the details to Ref.[4], which can be easily used to obtain the equations of motion in the case with the dilaton field.
10See Ref.[25]for the relevance of the Freund-Nambu theorem
to dense matter problems. There the problem of theω mass was not addressed.
By solving the coupled equations of motion, one can calculate the properties of a single skyrmion. Shown in TableIare the skyrmion properties obtained in dHLS-I and dHLS-II. We present the results with the dilaton parameters given in Eq.(34). For comparison, we show the results in HLS that has no dilaton field. By varying the dilaton mass we could also confirm that, the heavier the dilaton mass, the closer the results of dHLS-I and dHLS-II come to those of HLS, as anticipated.
The inclusion of the dilaton does indeed reduce the soliton mass, although not as much as needed. The mass reduction is found to be∼50 MeV and ∼90 MeV, respec-tively, for dHLS-I and dHLS-II.11 Given that the dilaton mass term itself increases the soliton mass by about the same amount in magnitude, we see that the attraction due to the dilaton coupling to other fields is twice the mass reduction, which is substantial. Here, the main contribution comes from the factor ðχ=fχÞ2 in the Lagrangian Lð2Þ, which is less than 1 in the central region of the skyrmion. On the other hand, in dHLS-I, the ω mass is reduced effectively by the same factor and provides more repulsion to the solution, as can be checked by the increase of the soliton mass from the hWZ terms. As for the N-Δ mass difference denoted byΔM in TableI, the dilaton causes its increase. It is not desirable but can be understood from the fact that ΔM is inversely proportional to the moment of
inertia with respect to the isospin rotation. With the dilaton field, the factorðχ=fχÞ2 inLð2Þ causes smaller moment of inertia that leads to a larger ΔM. On the other hand, the
skyrmion size is almost unaffected by the presence of the dilaton. The rms radius of the soliton evaluated by weighting the baryon number density, pffiffiffiffiffiffiffiffiffiffiffihr2iB, is almost unchanged by the incorporation of the dilaton, and the energy density weighted rms radiuspffiffiffiffiffiffiffiffiffiffiffihr2iE shows only a slight change. The breakdown of the soliton mass in each model is shown in Table I.
The profiles of the wave functions of I and dHLS-II are shown by dashed and dot-dashed lines, respectively, in Fig.1. For comparison, the wave functions of HLS are also given by solid lines, for which CðrÞ ¼ 1 and hence is not drawn. One can find that the wave functions FðrÞ, GðrÞ, and WðrÞ are almost unaffected by the presence of the dilaton field. This explains that the skyrmion size is almost unaffected by the dilaton field, as is verified in Table I.
However, the changes in WðrÞ show opposite behaviors in dHLS-I and dHLS-II. We can understand such a change in WðrÞ by the fact that the ω mass scales by the factor ðχ=fχÞ in dHLS-I, becoming effectively lighter in the
central region, while it is not scaled in dHLS-II. The wave function of the dilaton field CðrÞ illustrates that the scale symmetry—and consequently the chiral symmetry—is partially restored in the central region of the skyrmion, which is consistent with the chiral bag picture[28,29].
IV. DENSE SKYRMION MATTER IN dHLS-I AND dHLS-II
A. FCC skyrmion crystal
Although the effect of the dilaton on a single skyrmion is relatively minor, the dilaton plays a far more important role in dense matter made of such skyrmions. The dense skyrmion matter can be constructed by putting the sky-rmions onto the FCC crystal sites following the procedure outlined, for example, in Refs.[2,5,23]. To get the lowest energy configuration, the skyrmion at each lattice site should be arranged in such a way that the nearest skyrmions have the maximum attraction, for which the skyrmions at the closest sites should be relatively rotated in the isospin space by an angleπ about the axis perpendicular to the line joining them. This requires thatπðrÞ, ρμðrÞ, ωμðrÞ, and χðrÞ TABLE I.ffiffiffiffiffiffiffiffiffiffiffi Numerical results of the skyrmion properties. MsolandΔMð≡MΔ− MNÞ are in units of MeV, while
hr2i B p andpffiffiffiffiffiffiffiffiffiffiffihr2iEare in fm. ffiffiffiffiffiffiffiffiffiffiffi hr2i B p ffiffiffiffiffiffiffiffiffiffiffi hr2i E p ΔM Msol M Oðp2Þ sol M Oðp4Þ
sol Manomsol Mdilatonsol
HLS 0.43 0.59 522.8 1188.8 878.4 −125.1 435.4 0 dHLS-I 0.43 0.60 555.1 1138.0 746.2 −114.9 458.0 48.8 dHLS-II 0.41 0.58 636.0 1099.1 696.0 −117.1 431.4 89.0 0 −0.5 1 2 3 F(r) HLS dHLS-I dHLS-II −1.5 −1 −0.5 0 G(r) 0 0.5 1 1.5 r (fm) −0.8 −0.6 −0.4 −0.2 0 W(r) 0 0.5 1 1.5 2 r (fm) 0 0.2 −0.2 0.4 0.6 0.8 1 C(r)
FIG. 1 (color online). Profiles of the wave functions in HLS (solid lines), dHLS-I (dashed lines), and dHLS-II (dot-dashed lines). In HLS, CðrÞ ¼ 1 and it is not drawn here.
11In the minimal model, this reduction is about 50 MeV[27].
for the FCC crystal configuration12 should obey the periodic but distorted boundary conditions associated with the required symmetries with respect to the translation, reflection, fourfold rotations, and so on. (See Ref.[23]for details.)
The classical solutions for π, ρ, ω, and χ mesons satisfying the symmetries and carrying a specified baryon number per box can be obtained by applying the Fourier expansion method developed in Ref. [30]for the original Skyrme model, then generalized in Ref.[23]for the model with vector mesons and dilaton. Forπ, ρ, and ω fields, we use the convention of HLSðπ; ρ; ωÞ described in Ref. [5]. The isoscalar dilaton field χ could be expanded as
χðrÞ fχ ¼ X abc βχabccos πax L cos πby L cos πcz L ð40Þ with the expansion coefficientsβχabc with the same integer set ða; b; cÞ as that of the ω meson. Here, L is the half length of the edge of a single FCC box containing four skyrmions. The normal nuclear matter density n0¼ 0.17=fm3corresponds to the crystal size L∼ 1.43 fm.
The minimum energy configuration can be found numerically by taking the expansion coefficients as the variational variables. However, theω meson field needs a special treatment as in Ref. [23]. Since the ω meson provides a repulsive interaction and gives a positive definite contribution to the energy, a straightforward variational process always ends up with the trivial resultsω0¼ 0. It is nonetheless the correct solution to the equation of motion for the ω with the nonvanishing source term; viz.,
½−∂i∂iþ C2ðrÞm2ωω0ðrÞ ¼ SωðrÞ; ð41Þ where CðrÞ ¼ χðrÞ=fχ for dHLS-I 1 for dHLS-II: ð42Þ
Both in dHLS-I and dHLS-II, the source term, Sω in Eq. (41), comes from the hWZ terms,
Sω¼ −gωNc
32π2εijk½ðc1þ c2Þ~α⊥i·ð~α∥j× ~α∥kÞ
þ ðc1− c2Þ~α⊥i·ð~α⊥j×~α⊥kÞ
− 2c3fVij·~α⊥k− εijk∂ið~α∥j·~α⊥kÞg: ð43Þ
Thanks to the symmetries of the fields, the source term can be expanded in the Fourier series with the same set of the integersða; b; cÞ as those of ωðrÞ and can be written as
SωðrÞ ¼X abc γabccos πax L cos πby L cos πcz L : ð44Þ Then, the equation of motion for theω can be reduced to a linear matrix equation forβωabc as
X
a0b0c0
Dabc;a0b0c0βaω0b0c0 ¼ γabc; ð45Þ
where the matrix elements D−∂2i
abc;a0b0c0from the Laplacian∂2i
and Dm2ω
abc;a0b0c0 from the ω mass term in dHLS-II form
diagonal matrices, D−∂2i abc;a0b0c0 ¼ ða2þ b2þ c2Þ π L 2 δaa0δbb0δcc0; Dmabc;a2ω 0b0c0 ¼ mω2δaa0δbb0δcc0: ð46Þ
The matrix DC2m2ω from the space-dependentω mass term
with C2ðrÞ in dHLS-I is nondiagonal and its element has the form of DC2m2ω abc;a0b0c0 ¼ m2ω X a00;b00;c00 βC2 a00b00c00fa0a00afb0b00bfc0c00c; ð47Þ
with the Fourier expansion coefficientsβC2
abc for C2ðrÞ and
fa0a00a¼ 8 < : δa0a if a00¼ 0; δa00a if a0¼ 0; 1 2δa;a0a00 if a0a00≠ 0. ð48Þ Finally, the Fourier expansion coefficients βωabc can be obtained by multiplying the inverse matrix D−1 toγabc.
In Fig. 2, we present the obtained energy per baryon ðE=BÞ as a function of the crystal size L. The contributions fromLð2Þ,Lð4Þ, andLanom to E=B are also presented. The
results from dHLS-I and dHLS-II are shown by dashed and dot-dashed lines, respectively. For comparison, HLS results are also shown by solid lines. Besides the overall reduction in E=B due to the changes in the single skyrmion mass, as discussed in the previous section, we can see that, in the case of dHLS-I, nminwhere E=B has the minimum value is slightly moved to a lower value than that of HLS. In the case of dHLS-II, there is no noticeable change in nminbut E=B drops suddenly at a density denoted by nc≃ 4n0
whose position is given by the vertical solid lines in Fig.2, i.e., at L≃ 0.9 fm. As we will see later, at this density, not only does the overall average of the dilaton field vanish but alsoχðrÞ ¼ 0 in the whole space. In Fig.2(b), we can see explicitly the effect of the dilaton on the Lð2Þ, Lð4Þ, and Lanom contributions to E=B. One can see that they have
similar density dependence. It shows clearly that, as in the case of a single skyrmion, the dilaton field mainly affects the contributions from Lð2Þ. Again, this reflects that the dilaton couples only to the terms ofLð2Þ.
Figure 3 shows the space averaged quantities, hσi and hχi, as functions of the crystal size L. Here, σ is defined by 12In this paper, we deal with the skyrmion crystal only at the
U¼ ξ2¼ σ þ iτ · ϕπand the space averagehAi of quantity AðrÞ is hAi ¼ 1 Vbox Z box d3rAðrÞ; ð49Þ
where the integral is over a single FCC box with the volume Vbox¼ 8L3. The vanishing of hσi signals the skyrmion– half-skyrmion phase transition. In Fig.3(a)it is shown that the density of the half-skyrmion phase transition n1=2 is changed with the inclusion of the dilaton. Interestingly, they are opposite in dHLS-I and dHLS-II; in dHLS-I, n1=2 becomes slightly lower than that of HLS, while it becomes slightly higher in dHLS-II.
As can be seen in Fig.3(b), the dependence ofhχi on the crystal size at high density is completely different in dHLS-I and dHLS-II. In dHLS-I, as density increases,hχi decreases until the density approaches to about nmin, but after that it
begins to increase. Such an increase inhχi has been reported in the previous work of Ref.[23]. Once we accept that the scale symmetry is locked to the chiral symmetry, we expect hχi to decrease, not increase, as density increases. The main reason for this behavior comes from theχ2term that we have introduced into the second term inLð2Þ, which makes theω
mass scale withχ. The contribution of the hWZ term to E=B can be approximately expressed as
ðE=BÞanom ¼ 1 4 Z box d3r Z d3r0SωðrÞexpð−m ωjr − r0jÞ 4πjr − r0j Sωðr0Þ; ð50Þ
where mωis the“effective” ω mass. Note that the integration overr is restricted in the single FCC box but that over r0 is over all the space. In order to make ðE=BÞanom finite, the screening through a nonvanishingω mass is unavoidable.
The model based on dHLS-II, where ½Uð2ÞHLS sym-metry is broken down to ½SUð2Þ × Uð1ÞHLS, avoids the above-mentioned difficulty. We see in Fig. 3(b) that hχi smoothly decreases to, and beyond, n1=2and drops rapidly to zero when a higher density nc is reached.13
1000 1100 1200 1300 1400 E/B (MeV) HLS dHLS-I dHLS-II 0.5 1 1.5 2 2.5 L (fm) −200 0 200 400 600 800 1000 E/B (MeV) M sol (2) M sol (4) M sol (anom) (a) (b)
FIG. 2 (color online). (a) Energy per baryon E=B as a function of crystal size L. (b) Contributions fromLð2Þ,Lð4Þ, andLanomto
E=B. The results of dHLS-I and dHLS-II are given by the dashed and dot-dashed lines, respectively, while those of HLS are shown by solid lines for comparison. The vertical dotted line shows the position of normal nuclear density, i.e., L¼ 1.43 fm, and the vertical solid line shows that of the critical density nc that
corresponds to L≃ 0.9 fm. 0 0.2 0.4 0.6 0.8 1 〈 σ 〉 HLS dHLS-I dHLS-II 0.5 1 1.5 2 2.5 L (fm) 0 0.2 0.4 0.6 0.8 〈 χ 〉 / fχ (a) (b)
FIG. 3 (color online). (a) The expectation value hσi and (b)hχi=fχ as functions of crystal size L. The notations are the same as in Fig.2.
13As discussed in Ref. [31], an alternative way to get finite
ðE=BÞanom is to introduce a scale-dependent gω to weaken the
source itself. In Ref.[31], this was realized by multiplying the Sω term by the factorχ3so that the decrease in the effectiveω mass is accompanied by the decrease in the effective source. However, it is found that the weakening of theω coupling upsets the stability of the single skyrmion for a light dilaton that is needed for nuclear phenomenology. As a variation along this direction, one can endow an explicit density dependence in gω. In this case, there is no problem with the single skyrmion properties. This will be described in the discussion section.
B. In-medium properties of mesons
Once the skyrmion crystal is constructed, one can use it to study the in-medium properties of mesons as proposed in Refs. [2,32]. Taking the skyrmion crystal solution as background classical fields, we can interpret the fluctuating fields on top of it as the corresponding mesons in dense baryonic matter. For this purpose, we denote the minimum energy solutions as ξð0ÞðrÞ, ρað0Þμ , ωð0Þμ , and χð0Þ, and then introduce the fluctuating fields as
ξL;R¼ ~ξL;Rξð0ÞL;R; VðρÞ;aμ ¼ 1 2gρρ að0Þ μ þ 12gρ~ρaμ; VðωÞμ ¼ 1 2gωω ð0Þ μ þ 1 2gω~ωμ; χ ¼ χð0Þþ~χ; ð51Þ
where ~ξ†L¼ ~ξR¼ ~ξ ¼ expðiτa~πa=2fπÞ,~ρaμ, ~ωμ, and~χ stand
for the corresponding fluctuating fields. In Eq.(51), gρand gω are the medium modified HLS gauge couplings of theρ and ω mesons, respectively. It is worth noting that the decomposition given in Eq.(51)can easily keep the HLS of the matter in terms of the expansion of the quantum fluctuations by imposing that the fluctuations transform homogeneously under the HLS, but the matter fields transform the same as their corresponding original quantities in HLS. By substituting the fields in Eq. (51)
into the dHLS Lagrangian, one can obtain the medium modified one.
To define the pion decay constant in the skyrmion matter, we consider the axial-vector current correlator
iGab μνðpÞ ¼ i
Z
d4xeip·xh0∣TJa
5μðxÞJb5νð0Þ∣0i: ð52Þ
This correlator can be evaluated from the medium modified Lagrangian by introducing the corresponding external source by gauging the chiral symmetry, i.e., substituting the covariant derivative defined in Eq.(5) with
DμξL¼ ð∂μ− iVμÞξLþ iξL;RLμ;
DμξR¼ ð∂μ− iVμÞξRþ iξL;RRμ; ð53Þ
whereLμandRμare introduced as the gauge fields of the chiral symmetry. The external source of the axial-vector current Jμ5 is a combination ðRμ− LμÞ=2.
In the present calculation, we do not consider the contributions from the loop diagrams of the fluctuation fields to the correlator of Eq.(52). Therefore, as illustrated in Fig. 4, there are three types of contributions: (i) the contact diagram, (ii) the pion exchange diagram, and (iii) the ρ exchange diagram. In the present evaluation of the correlator, we only consider the matter effect from
ξð0ÞL;R andχð0Þbut leave a complete calculation, including
derivative on them, to our future publication. In such an approximation, the three types of contributions are expressed as ðiÞ∶ if2 πgμνδab χ2 ð0Þ f2χ 1 þ1 − a 2 1 −2 3ϕ2π − 1 ; ðiiÞ∶ − if2 π pμpν p2 δ abχ 2 ð0Þ f2χ 1 −2 3ϕ2π − 1 ; ðiiiÞ∶ iδabχ 2 ð0Þ f2χ a2g2f4π p2−χ 2 ð0Þ f2χ m 2 ρ gμν−χp2μpν ð0Þ f2χ m 2 ρ ×χ 2 ð0Þ f2χ 1 −2 3ϕ2π − 1 : ð54Þ
Summing over the above three types of contributions, one concludes that, to the leading order of the p2=ðχ2ð0Þm2ρ=f2χÞ expansion, the axial-vector current correlator(52)is gauge invariant and therefore can be decomposed into the longitudinal and transverse parts as
Gab
μνðpÞ ¼ δab½PTμνGTðpÞ þ PLμνGLðpÞ; ð55Þ
where the polarization tensors PL;T are defined as
PTμν¼ gμi δij− pipj jpj2 gjν; PLμν¼ − gμν−pμpν p2 − PTμν: ð56Þ
We next define the medium modified pion decay constant through the longitudinal component in the low energy limit
f2π ≡ − lim p0→0 GLðp0;p ¼ 0Þ ¼ f2 π χ2 ð0Þ f2χ 1 −2 3ð1 − σ2ð0ÞÞ ; ð57Þ
where the intrinsic density dependence is brought in by the minimal energy solution ðχð0Þ=fχÞ2 and σ2ð0Þ, and the relation σ2ð0Þþ ϕ2π¼ 1 has been used. Note that, because of the rho meson exchange effect, the medium modified fπ is independent of the HLS parameter a.
FIG. 4. Three types of contributions to the correlator of Eq.(52): (i) the contact diagram, (ii) the pion exchange diagram, and (iii) the ρ exchange diagram. Shaded blobs stand for the skyrmion matter interaction vertices.
In our present approach, since only the Oðp2Þ terms of
HLS are considered, we should have
gρ¼ gρ ð58Þ
for the normalization of theρ meson field in medium. Thus, because of the dilaton compensator, theρ meson mass is modified to be m2ρ ¼χ 2 ð0Þ f2χ m2ρ: ð59Þ
As for the ω meson mass, dHLS-I and dHLS-II lead to different results: m2ω ¼ 8 < : χ2 ð0Þ f2χ m2ω for dHLS-I; m2ω for dHLS-II ð60Þ From Eqs. (59) and (60) one may think that the BR scaling for the ρ and ω meson masses is reproduced in dHLS-I[12]. However, as will be shown below, since hχi increases with increasing density above n1=2, the masses increase so they are not consistent with the BR scaling for n≥ n1=2. Also the pion decay constant scales differently from that of the vector meson masses.
Plotted in Fig.5are fπ=fπ and mρ=mρthat show their dependence on the crystal size. Since mω=mω is equal to mρ=mρin dHLS-I or it does not scale in dHLS-II, we do not show it here. As for fπ=fπ, for density up to ∼n1=2, the scaling behavior is mainly governed byσð0Þ, so that both in dHLS-I and in dHLS-II it decreases smoothly down to fπ=fπ∼ 2=3.
For density larger than n1=2, the scaling behavior is governed byχ2ð0Þ=f2χ, for which dHLS-I and dHLS-II yield different results. In both cases, fπ=fπ stays ∼2=3 for a while. Then, after nminit starts to increase in dHLS-I, but it
goes down at higher density and then drops to zero at ncin dHLS-II. The in-medium ρ meson mass mρ scales only with
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hχ2
ð0Þ=f2χi
q
that is similar tohχð0Þ=fχi, so it decreases monotonically until∼nmin, after which it starts to increase
in dHLS-I, although it will ultimately drop to zero at ncin
dHLS-II.
The situation in dHLS-II is much simpler and appealing over all the range of density. The quantities tied to chiral symmetry, fπ=fπand mρ=mρ, do vanish at nc, showing that
chiral symmetry is restored and the vector manifestation is realized. It is intriguing that the VM is realized at the expense of breaking the ½Uð2ÞHLS symmetry in medium and letting the ω meson mass remain unscaled.
C. The half-skyrmion phase
The half-skyrmion phase exhibits some unusual proper-ties of hadrons. This may be indicative of a non-Fermi
liquid structure mentioned above[21]. As one can see in Fig.5, in the phase for n≥ n1=2,hσi ¼ 0 but fπ≠ 0. At the edge of the half-skyrmion phase, say, at nc, the
half-skyrmion phase presumably transits to the Wigner phase withhχi ¼ 0 and fπ¼ 0. In the Wigner phase the chiral symmetry is restored.
A question that arises is how to formulate the vector manifestation starting from the half-skyrmion phase. In matching the HLS and QCD correlators in arriving at the VM[9,13], the conditionh¯qqi → 0 plays a key role. Now in the half-skyrmion phase,h¯qqi ¼ 0 but the pion decay constant is not zero. So we see that the VM cannot be realized in the half-skyrmion phase. Furthermore, since gðmρÞ ≠ 0 because of m2
ρ¼ aðmρÞg2ρðmρÞf2πðmρÞ ≠ 0, the
Georgi’s “vector realization” of chiral symmetry [33,34]
cannot be arrived at.
Now let us approach the Wigner phase from the half-skyrmion phase. At density nc, the effective pion decay
constant and the ρ meson mass take the values
mρðncÞ ¼ 0; fπðmρðncÞÞ ¼ fπð0Þ ¼ 0; ð61Þ
which can be regarded as the conditions to realize VM in medium. In this sense, we can say that the VM of Eq.(1)
could be realized in the model of dHLS-II.
It has been discussed in the literature[31,35]that there is a possible pseudo–gap phase in QCD in which quarks condensate and acquire constituent mass, but chiral
0 0.2 0.4 0.6 0.8 1 f π ∗ / f π HLS dHLS-I dHLS-II 0.5 1 1.5 2 2.5 L (fm) 0 0.2 0.4 0.6 0.8 1 m ρ ∗ / m ρ (b) (a)
FIG. 5 (color online). The dependence of (a) fπ=fπ and (b) mρ=mρ on the crystal size L. The notations are the same as in Fig.2.
symmetry is not broken because the condensate phase is completely disordered. This situation is very similar to what happens in the half-skyrmion phase where the space average of quark-antiquark condensate vanishes and theρ is still massive, indicating that the quark acquires a constituent mass. Thus if the order parameter of QCD were interpreted as the space averaged quark-antiquark condensate as in Refs.[31,35], the half-skyrmion phase can be taken as the pseudo–gap phase. It could also be “quarkyonic.”14
Note that, in the half-skyrmion phase, the quark condensate is locally nonzero.
D. In-medium baryon properties
Substituting the inputs fπ, mρ, and mω with the corre-sponding medium modified fπ, mρ, and mωinto the master formula(26), one can obtain the in-medium LECs of HLS. Using these in-medium LECs we can then calculate the intrinsic density-dependent nucleon mass. We plot in Fig.6
the effective mass Msol to see its density dependence. From Fig. 6, we see that the soliton mass, which is identified as the nucleon mass in the large Nc limit, can be parametrized as
mN¼ m0þ Δðh¯qqiÞ; ð62Þ whereΔ is the part of the nucleon mass arising from h¯qqi that vanishes at n¼ n1=2 and m0 is the chiral-invariant mass, reminiscent of what figures in the parity doublet picture of baryons[11].
We next make a parametrization of the scalings of fπ, mρ, and mNfrom our results shown in Figs.5and6. Such a parametrization would make some results of the present work easier to apply, for example, to the nuclear force and the EoS of nuclear matter[21]. Here we only consider the results from dHLS-II. Note that although fπand mN scale similarly, mρscales in a different way.
We can roughly parametrize the medium-modified pion decay constant fπillustrated in Fig.5(a)and the in-medium nucleon mass mN illustrated in Fig. 6as
mN mN ≃fπ fπ≃ 8 < : 1 1þ0.6ðn=n0Þ2 for n < n1=2; 0.63 for n1=2< n < nc; 0 for n > nc: ð63Þ
For the in-medium vector meson mass mρin Fig.5(b)we parametrize it as mρ mρ≃ ( 1 1þ0.4ðn=ncÞ2 for n < nc; 0 for n > nc: ð64Þ
V. SUMMARY OF THE RESULTS
The series of work done with multi-skyrmions obtained from a chiral Lagrangian with vector mesons incorporated as hidden gauge fields to simulate dense baryonic matter revealed a number of features that were not observed in chiral models without vector mesons. The model used in the present work is based on the Lagrangian written up to Oðp4Þ in the chiral expansion, including the pion, the ρ
meson, and theω meson, and is parameter free thanks to master formulas derived from the 5D holographic QCD action that arises from gauge-gravity duality of hQCD, as well as from dimensional deconstruction starting from SUð2ÞL× SUð2ÞR current algebra. This Lagrangian is
considered to be as close as one can hope to reach the large Nc limit of QCD properly.
To repeat the most remarkable results:
(1) The isosinglet vector mesonω plays a crucial role in the structure of both the elementary nucleon and multinucleon systems. Given that theω meson is in the topological term encoding anomaly, it cannot be properly, if at all, captured in models that have no explicit ω degree of freedom such as the famous Skyrme model or chiral perturbation theory. (2) The density n1=2 at which the skyrmion
–half-skyrmion transition, a generic feature of all the skyrmion models on crystal, takes place is found to be not far from the equilibrium nuclear matter density n0∼ 0.17 fm−3. It is therefore testable ex-perimentally, such as through the medium modified kaon mass[36]and nuclear tensor force[37](for a recent review, see, e.g., Ref.[38]). Without theρ and ω fields, the transition takes place at much too low a density to be compatible with what is accurately known in normal nuclear matter, and without the hWZ term—i.e., without the ω meson—it comes much too high to be relevant to nature.
0.5 1 1.5 2 2.5 L (fm) 400 600 800 1000 1200 M sol * (MeV) HLS dHLS-I dHLS-II
FIG. 6 (color online). In-medium modified skyrmion mass as a function of L. The notations are the same as in Fig.2.
14But we know that at least on the crystal lattice, the quark
(3) In medium, the effective pion decay constant fπ encoding the intrinsic density dependence is found to drop smoothly, roughly in consistency with chiral perturbation theory, to the density n1=2, but stops dropping at n1=2and remains constant∼ð60%–80%Þ of the free-space value in the half-skyrmion phase. (4) The in-medium nucleon mass mN tracks closely the in-medium pion decay constant fπ(multiplied by a scale-invariant factor proportional to ffiffiffiffiffiffiNc
p
) which indicates that the large Nc dominance holds in
medium as it does in free space and stays constant ∼ð60%–80%Þ of the free-space value in the half-skyrmion phase for n≥ n1=2. Given thath¯qqi¼ 0 in the half-skyrmion phase, the constant nucleon mass must be a chirally invariant term in the Lagrangian. This resembles the chiral-invariant m0 term in the parity-doublet baryon model. In our calculation, such a term is not explicitly present in the Lagrangian with which the skyrmion crystal is constructed. Therefore, it could very well be a symmetry“emergent" from many-body correlations different from what Glozman interprets as an in-trinsic property of QCD in Ref. [39].
The items 3 and 4 get support from independent analyses based on the one-loop RG flow with baryon HLS (BHLS) and mean-field approximation with dilaton-implemented BHLS[24]. This suggests that the qualitative structure of the half-skyrmion phase is correct. However, as pointed out in the present work, there is a tension with the VM and BR scaling. This is because, since the pion decay constant and theρ meson mass stay constant and do not tend to the VM fixed point, one cannot make the matching of the HLS correlators to those of QCD crucial to describe chiral restoration at a certain high density nc.
In this paper, we remove the obstacle to the VM found in the HLS crystal by the dilaton field associated with the spontaneous breaking of conformal symmetry. We found that in order for the dilaton to retain the VM feature or BR scaling, the ½Uð2ÞHLS symmetry for the ρ and ω which seems to hold in matter-free space has to be broken in medium. With the symmetry breaking induced in the vector meson mass rather than in the gauge coupling as was done in Ref. [24], all four features mentioned above were retained and, in addition, the tension with the VM present in the HLS model (without the dilaton) could be removed. We should point out that there are some caveats to the “good” results mentioned above. (1) The energy of the system is minimized at a larger density than the known equilibrium density n0with a binding energy much larger than the empirical value. This is not surprising since we have here a large Nctheory. In standard nuclear many-body
theory anchored on effective field theory, a similar over-binding and higher saturation density are obtained unless one introduces three- and multibody forces (see, e.g., Ref. [40]). Whether this “higher-order” effect is encoded
in the crystal calculation needs to be clarified. (2) In all cases considered, the density n1=2 comes below the empirical value of n0, with the HLS-II giving n1=2 close to n0. From the phenomenology discussed in Ref.[41], the density n1=2most likely relevant to nature should be≲2n0. However, it seems that there will be no difficulty even if n1=2 comes close to but above n0. (3) The half-skyrmion structure is a classical picture, already present in the skyrmion description of mass number 4 (see Ref. [42]) and will surely be modified by quantum effects.
VI. FURTHER DISCUSSIONS
It is interesting to compare what we have found in this paper to what have been seen in other developments, which are closely related to each other.
(i) Phenomenologically relevant is the application of the skyrmion–half-skyrmion topology change to an effective nuclear field theory description of the EoS for compact stars [41]. Using the notion that top-ology change can be“translated” into the parameter change of an effective Lagrangian, here in the form of the“intrinsic density dependence” defined above, an effective nuclear Lagrangian was constructed by means of renormalization group equations and applied to calculating a high-order nuclear many-body problem. The principal observation there was that in order to correctly describe nuclear matter and then extrapolate to higher density, it was essential that the effective nucleon mass drop smoothly to ∼0.8mN up to density∼2n0 and then
stay constant up to the density∼5.5n0predicted to be present in the interior of a massive neutron star and that theωNN coupling be more or less unscaled in the density regime involved. This is roughly the feature obtained in dHLS-II.
(ii) Suppose we follow the development made in Ref.[24]
using the mean-field approximation in dHLS-I, sup-plemented with explicit baryon degrees of freedom and ½Uð2ÞHLS symmetry breaking in the gauge coupling constants instead of in the vector meson masses. In the dHLS-Iðπ; ρ; ωÞ crystal calculation, the corresponding procedure would be to replace gω with a density-dependent one in the form of
gω→ gω 1
1 þ Bðn=n0Þ; ð65Þ
where n0 is the normal nucleon density and B is a parameter. We refer to this model as dHLS-Ia and present the obtained results for the per-skyrmion energy and Msol in Fig. 7 for two cases with B¼ 0.10 and 0.22, respectively. The dilaton mass is taken to be mχ ¼ 720 MeV, as before. Our result shows that to arrive at the minimum of E=B, a smaller parameter B is preferred. It clearly shows that there is
a density region after n1=2in which the nucleon mass is nearly density independent. This agrees with the observation made in Ref. [24]. Note that because of the density-dependent coupling gω, n1=2is pushed to a higher density by the dilaton.
(iii) In the HLS calculation, we find that, in the half-skyrmion phase, the space average hσi ¼ 0, but hσ2i ≠ 0 although it is a small density-dependent
quantity. This observation might indicate that, although the space average quarkonia condensate vanishes in the half-skyrmion phase, the space average tetraquark condensate does not. Probably this is the first example which realizes the chiral symmetry breaking pattern SUðNfÞL× SUðNfÞR→
SUðNfÞV× ZðNfÞAproposed in Ref.[43]emergent
in dense baryonic matter, whose features of the thermodynamic quantities and hadron mass spectra of this phase in the two flavor case are explored in Ref. [44]. If one assumes the Gell-Mann–Oakes– Renner relation holds in the half-skyrmion phase in the form fπ2mπ2¼ Dhσ2i, with D a density-independent constant, then since hσ2i drops to a small value while fπremains more or less constant, one should expect mπ should accordingly decrease fast in the half-skyrmion phase in the real world where the pion mass is nonzero. This would imply that the contribution from the pion exchange to the nuclear tensor forces will be enhanced for density
n≥ n1=2and hence will increase the net tensor-force attraction, as is clear from the finding in Ref.[41]. As was pointed out a long time ago by Pandhar-ipande and Smith [45], such an enhanced tensor force could lead to a p-waveπ0-condensed neutron solid at high density in compact stars.
(iv) A remarkable feature that has been uncovered in all models anchored on HLS is that at a certain density above the normal nuclear matter density, the effec-tive nucleon mass mN stops being dependent on the chiral condensate, saturating at ≳60% of the free-space mass, and then stays constant until quark deconfinement. We note that this is reminiscent of the chiral-invariant mass m0 posited in the parity-doublet baryon model. The mass m0 could be an intrinsic quantity of QCD proper in the sense suggested in Ref.[39]. However, the parity-doublet baryon model is an effective theory. Now in our treatment, the chiral-invariant mass—that breaks explicitly conformal invariance—is not put in ab initio in the Lagrangian. Therefore, it is plausible that it rather reflects an emergent symmetry due to HLS skyrmion interactions, and not an intrinsic one. We note that since what we are looking at is a process of“unbreaking symmetry” by density, this indicates a subtle mechanism by which the nucleon mass could have been generated.
ACKNOWLEDGMENTS
We are grateful to APCTP for supporting the APCTP-WCU Focus program where this work was initiated. This work was completed while three of us (Y.-L. M., B.-Y. P., and M. R.) were visiting the Rare Isotope Science Project (RISP) funded by the Ministry of Science, ICT, and Future Planning (MSIP) and National Research Foundation (NRF) of Korea. We are grateful to Youngman Kim for making this visit feasible and Y.-L. M. acknowledges the support from the RISP. The work of Y.-L. M. and M. H. was supported in part by a Grant-in-Aid for Scientific Research on Innovative Areas (No. 2104)“Quest on New Hadrons with Variety of Flavors” from MEXT. Y.-L. M. was supported in part by the National Science Foundation of China (NSFC) under Grant No. 10905060. The work of M. H. was partially supported by the Grant-in-Aid for Nagoya University Global COE Program“Quest for Fundamental Principles in the Universe: From Particles to the Solar System and the Cosmos” from MEXT, the JSPS Grant-in-Aid for Scientific Research (S) No. 22224003 and (c) No. 24540266. The work of H. K. L. and M. R. was partially supported by the WCU project of Korean Ministry of Education, Science and Technology (R33-2008-000-10087-0). Y. O. was supported by the Basic Science Research Program through the National Research Foundation of Korea under Grant No. NRF-2013R1A1A2A10007294. B.-Y. P. was supported by the research fund of Chungnam National University.
600 800 1000 1200 1400 1600 E/B (MeV) HLS dHLS-Ia (B = 0.10) dHLS-Ia (B = 0.22) 0.5 1 1.5 2 2.5 L (fm) 400 600 800 1000 1200 M sol * (MeV)
FIG. 7 (color online). E=B and Msol in dHLS-Ia with mχ ¼ 720 MeV.
